## Euclid: Elements |

Ancient Greek text and translations presented by
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Euclid |
Euclid is the author of The Elements, the
definitive work on classical geometry, which today is
named after him: we call it “Euclidean geometry”, to
distinguish it from other, non-Euclidean geometries that
were invented in the 19th century.Nearly nothing is known of Euclid, except that he lived in Alexandria, Egypt, during the reign of Ptolemy I (323 BC – 283 BC). He was active at the great Library of Alexandria, and may have studied at Plato’s Academy in Athens, Greece. The Elements consist of 13 books, some of which deal with
domains of mathematics that today we call algebra and
number theory, but the ancient Greeks treated these
subjects always from a geometrical perspective. The
Elements treat such important topics in |

In algebra and number theory,
topics include divisibility, the prime numbers, greatest common
divisor, least common multiple, prime factorization, proof of the
infinity of primes, perfect numbers, geometric sequences, sums of
geometric series, irrational numbers, and more.Finally, in Besides being a basic reference for geometrical
and number-theoretical knowledge, another, most important
contribution of the Elements is that it exemplified the Euclid’s proofs are not error-free, because
in some cases he makes use of propositions that appear “obvious”,
but which he never stated (or proved) explicitly. This, however,
is unavoidable, as anyone who has worked in automated theorem
proving knows: if one wants to provide a complete proof of
anything but the simplest statements, the true number of
propositions on which the proof must rest explodes to proportions
unmanageable by the human mind — hence only machines can keep
track of some proofs in their entirety. Euclid used shortcuts, as
every schoolteacher who teaches geometry inevitably must do. |

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